First of all, remember that the US is a very large and populated place. Your home country of New Zealand is only about the size and population of an average state in the US. Obviously, people in different areas are going to need different things, and different laws - for example, Nevada (a highly desert state) might need strict water usage laws, while those same laws would be quite silly in Minnesota, Land of 10,000 lakes.
Also note the historical influence - after we kicked out the British, each of the former colonial governments had significant power, more than the States do today; they gave up some of this power to form the Union we are in now. Also, the Founding Fathers of the US were quite paranoid about government tyranny; that is why the US government is full of checks and balances, and having states & the federal government competing to supposed to reduce this effect.
As for your specific questions:
A) Generally, the Federal government prints the currency, maintains foreign relations/declares wars, regulates interstate and international trade, has jurisdiction over certain crimes (especially those commited by someone crossing state lines), does some large scale infrastructure programs like the Interstate highway, and has managed to run big programs like Medicare/Medicade and Social Security. I suggest reading the American Constitution (don’t worry, it is not that long) for a more detailed view of what powers the federal government has.
The States usually handle local crime/justice, have their own social welfare programs, build the streets, run the public schools & universities. and such. The States also have their own military forces, collectively known as the National Guard, that can be called up when need for state emergencies, or called up by the federal government to fight in wars.
B)Each State does have its own soverign powers, and thus don’t like it when the Fed infringes on them, much like various countries don’t like others threating embargoes to get them to change their domestic policies. This is because the people of a state may have interest that are opposed by the federal government.
For example, in the US, the legal age to consume alcohol is 21. Now, the citizens of several states in the US would like to lower the legal age in those states, but they can’t, due the backhanded influence of the federal government. You see, their is a federal tax on gasoline, that the Federal government collects, then gives to various states to fund road maintance. If those states lowered their drinking ages, the amount of funding they would get from that tax would be reduced, yet the citizens of that state would still have to pay the gasoline tax.
C) The States are United in more than than foreign policy - the federal government maintains a uniform set of Interstate trade regualations, for example.
D) Anything can happen, but I find it unlikely that the US will be broken up anytime in the next century or two. In the past, several states tried to break away from the Union, leading to the whole mess that was the American Civil War. No state could leave unless the other states wanted them to. And I doubt any state would really want to leave - the benifits of being in the Union generally outweight the cost.